Talk:Arithmetic function

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I changed the article back to the previous version for the following reasons:

  • It used N for the positive integers, but we generally use that symbol for the non-negative integers, see natural number.
//1 Z+≡N. Non-negative are (or should be) denoted Z*. (Z*=0+N)
In Wikipedia, N stands for the non-negative integers, as natural number explains. We have to use consistent conventions, even if they are arbitrary.AxelBoldt
//2 I am not quite shure if above is really so well explained in subsistent wikipedian article natural number. About consistent conventions I won't argue - according to Ribenboim (1996) this matter about what natural numbers should be is not so important and it is dependent on the usage of the definitons. Famous French "group mathematician" Bourbaki and many more used 0 as natural number like Wikipedia. Others don't. For instance Möbius function μ(0) is trivial and it does not show any real properties of the function itself. I am now confused even more, but I would like to use consistent stuffs in math still...
Yes, what counts as a "natural number", and what is denoted by N, is a matter of convention. There are two competing coventions: number theorists reserve "natural number" and N for "positive integer", Bourbaki, set theorists and computer scientists reserve "natural number" and N for "non-negative integer". At one point, we in Wikipedia decided to follow the second convention. If that is not clear from natural number, we have to clarify that entry. AxelBoldt
//3 Would it be OK to add somewhere in natural number a simple table like this (non-Wikipedian conventions!):
integers {-∞...,-2,-1,0,1,2,...,+∞} Z
positive integers {1,2,...,+∞} Z+ ≡ N
non-negative integers {0,1,2,...,+∞} Z* ≡ 0+N
non-positive integers {-∞...,-2,-1,0} /*no designation */ ≡ Z-N
negative integers {-∞...,-2,-1} Z- ≡ Z-Z*
No, that would not be ok, because then people could think that Wikipedia uses N for positive integers, which it doesn't. I will add symbols to number. AxelBoldt
  • No need for colors in formulas.
//1 Red color was mentioned for unshure statement lately when cleared up put out.
You used green for a function arrow.AxelBoldt
//2 I saw two colors in formulas (blue and magenta) like this:
ln x e.
Why other colors are so scandalous? Where can I find that I can't use colors in math formulas? We should create something like /mathematics/how to write formuals, etc. I think I am violating nothing here...
  • The definition of Ψ didn't make sense: in one equation the function Ψ took two arguments, in the other it took only one. Furthermore, this does not seem to be an accepted use of the term "arithmetic function".
//1 DNA (Do not agree) Number of arguments are obviously proper property of this function. Another definition won't hurt noone. It would make him to think a little bit.
A function either takes two arguments or one. Your definition did not make sense, because it mixed the two. It didn't specify what the domain and codomain of Ψ is.AxelBoldt
//2 Says who? A god Math Zeus a.k.a. Axel?
I signed my statement, so you know who said it. Why play rhethorical games like this? AxelBoldt
//3 I think above is not true. In the first example function Ψ(m+n) has two parameters m and n if we watch that before adding them and it has one argument, let us say s=m+n. This is OK for me. In the second term of definition I guess it was not well understood. I think (we should check this) that Ψ(m,n) should be written as Ψ(m|n) or as pascalike Ψ(mDIVn). In this way function Ψ( X ) has the same number of parameters, where X can be whatever. About games. I didn't choose them. I advise you to check your own corrections of articles of the other authors. See bellow for the same matter on mathematical insignificance.
  • "Arithmetical function" was not explained.
//1 See article.
//2 I tried to explain that but obviously that was not explanation. Don't know what to do futher on. I saw in many ways that someone uses two different terms arithmetic(al) functions but I do not want to impose my opinions to anyone. And I have rights on my own opinions, do I?

AxelBoldt, Thursday, April 18, 2002

//1 How to move from the NPOV in this way?
//2 It is very easy to find 1001 reasons to revert one extended (not necessarily correct) article to previous state. --XiJam [2002.04.18] 4 Thursday (0).

I removed this again:

Another definition of arithmetic function is a function with exactly two compositional properties:

Ψ(m+n) = Ψ[Ψ(m)+Ψ(n)] /composition with addition/
Ψ(m,n) = Ψ[Ψ(m)Ψ(n)] /composition with parameter/

In the first equation, Ψ is used as a function of a single argument, in the second equation it is used as a function of two arguments. This definition is mathematically meaningless. AxelBoldt

XJam, where did you find these two lines involving Ψ? Maybe I can clarify it from the source. AxelBoldt

//1 At Eric's "cooking's place" http://mathworld.wolfram.com/ArithmeticFunction.html. We better go and kick his ass a little bit, ha, ha. That definition is really quite tedious, so I don't believe it would help you very much. I'll go check somewhere else.

I removed this:

Note: Arithmetic function should not be confused with a function sometimes called arithmetical function which is in fact integer function (f : N → N).

There is no such distinction between "arithmetical" and "arithmetic" functions in the literature. Furthermore, the above arithmetical functions also qualify as arithmetic functions of course. AxelBoldt

//1 No comment. --XJamΨ [2001.04.19] 5 Friday (0).
AO --XJamτ [2001.04.19] 5 Friday (1st ed).

[edit] arithmetic vs arithmetical

please, if concensus is reached could this be clarified in the article ?

  • The page's title is "arithmetic f." (and that where most pages link to).
  • The first paragraph (the only which is displayed with [navigation popup]s e.g.) only defines "number theoretic f.",
  • The second paragraph defines "arithmetical f."
  • The rest of the page only speaks of "arithmetic f."

— MFH:Talk 17:43, 2 April 2007 (UTC)